From: Jim Pryor Date: Tue, 2 Nov 2010 12:10:31 +0000 (-0400) Subject: cat theory: different bold X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=8076acb29d9f26cb505398e08d79bf5992584fc7 cat theory: different bold Signed-off-by: Jim Pryor --- diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index aef5538a..0f84bb28 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -40,7 +40,7 @@ Categories ---------- A **category** is a generalization of a monoid. A category consists of a class of **elements**, and a class of **morphisms** between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a single morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension." -When a morphism `f` in category **C** has source `C1` and target `C2`, we'll write `f:C1->C2`. +When a morphism `f` in category C has source `C1` and target `C2`, we'll write `f:C1->C2`. To have a category, the elements and morphisms have to satisfy some constraints: @@ -71,18 +71,18 @@ Some examples of categories are: Functors -------- -A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category **C** to category **D** must: +A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category C to category D must:
```-	(i) associate with every element C1 of **C** an element F(C1) of **D**
-	(ii) associate with every morphism f:C1->C2 of **C** a morphism F(f):F(C1)->F(C2) of **D**
-	(iii) "preserve identity", that is, for every element C1 of **C**: F of C1's identity morphism in **C** must be the identity morphism of F(C1) in **D**: F(1C1) = 1F(C1).
-	(iv) "distribute over composition", that is for any morphisms f and g in **C**: F(g o f) = F(g) o F(f)
+	(i) associate with every element C1 of C an element F(C1) of D
+	(ii) associate with every morphism f:C1->C2 of C a morphism F(f):F(C1)->F(C2) of D
+	(iii) "preserve identity", that is, for every element C1 of C: F of C1's identity morphism in C must be the identity morphism of F(C1) in D: F(1C1) = 1F(C1).
+	(iv) "distribute over composition", that is for any morphisms f and g in C: F(g o f) = F(g) o F(f)
```